Forced linear oscillators and the dynamics of euclidean group extensions

Research output: Contribution to journalArticlepeer-review


We study the generic dynamical behaviour of skew-product extensions generated by cocycles arising from equations of forced linear oscillators of special form. This work extends our earlier work on cocycles into compact Lie groups arising from differential equations of special form, (cf. [21]), to the case of non-compact fiber groups of Euclidean type. The earlier techniques do not work in the non-compact case. In the non-compact case one of the main obstacle is the lack of 'recurrence'. Thus, our approach to studying Euclidean group extensions is: (i) first, to use a 'twisted version' of the so called 'conjugation approximation method' and then (ii) to use 'geometric-control theoretic methods' developed in our earlier work (cf. [20] and [21]). Even then, our arguments only work for base flows that admit a global Poincaé section, (e.g. for the irrational rotation flows on tori and for certain nil flows). We apply these results to study generic spectral behaviour of the forced quantum harmonic oscillator with time dependent stationary force restricted to satisfy given constraints.

Original languageEnglish (US)
Pages (from-to)1201-1234
Number of pages34
JournalDiscrete and Continuous Dynamical Systems - Series S
Issue number4
StatePublished - Aug 2016

All Science Journal Classification (ASJC) codes

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


  • Cocycles into Euclidean groups
  • Ergodic skew products
  • Forced linear oscillators


Dive into the research topics of 'Forced linear oscillators and the dynamics of euclidean group extensions'. Together they form a unique fingerprint.

Cite this